#94973

order	512
nilpotency class	3
isoclinism	isoclinic class of #94972
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,6,4,5,7,8)(9,10,12,15,13,14,11,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,27)(28,31)(29,32);(10,14)(11,12)(17,19)(18,23,25,32)(20,29)(21,27)(22,28)(24,26,30,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=gh,e^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cg,e^b=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239977, #72257821
maximal normal subgroup of	#72239977, #72257821
maximal quotient of	#72240004

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2169);